This reminds me of 3.5, when Eberron released the Warforged, and immediately I recall some hubbub about Warforged Druids and not only whether they should be able to wild shape at all, but what they should be able to wild shape into, as a Construct-adjacent race turning into a flesh and blood animal seemed very odd (magic can, of course, do this sort of thing, at least as long as Polymorph Any Object has been in the game, but a lot of people I've found seem to think that what arcane magic can do is not thematic for Druids- never no mind Shapechange is a Druid spell).
While 3e had fairly specific rules about what abilities are lost/gained with wild shape, 5e does run into a problem since races are a mix of inherent and trained abilities (though this is changing as time goes on).
While Fleet of Foot doesn't specify that it's inherent to one's form, the argument you would have to make comes down to this: are Wood Elves faster than other races because of being Wood Elves, or do all Wood Elves learn to run really fast as a result of their lifestyle?
If it's the first, obviously, Wild Shape would make you lose it. If it's the second, it would not, but it would beg the question why anybody couldn't receive training to run really fast- since unlike the fast movement of a Monk or Barbarian, which any race can acquire, Fleet of Foot allows Wood Elves to be faster than other Monks or Barbarians.
Like so much in D&D, the answer is going to be "ask your DM", as you'll find people on both sides of the fence.